This article covers the essential skills of approximation, scientific notation, and error analysis – a recurring topic in DSE Paper 1 Section A(1) worth 3–5 marks. You will learn how to round numbers to a given number of significant figures or decimal places, how to express very large or very small numbers using scientific notation, and how to calculate absolute error, relative error, and percentage error. The article includes step-by-step worked examples, DSE exam techniques, practice questions, and a full set of homework with solutions. These are easy marks that every candidate should secure.
By the end of this article, you should be able to:
Approximation and errors appear every year in DSE Paper 1 Section A(1). Questions may ask you to:
These are straightforward marks if you know the rules. Don't lose them!
Questions on approximation and errors often appear in Section A(1) as short-answer questions worth 2–3 marks each. They are designed to test your basic numeracy skills.
Rounding to a given number of decimal places means keeping a certain number of digits after the decimal point.
| Number | Round to 1 d.p. | Round to 2 d.p. | Round to 3 d.p. |
|---|---|---|---|
| \( 3.14159 \) | \( 3.1 \) | \( 3.14 \) | \( 3.142 \) |
| \( 2.71828 \) | \( 2.7 \) | \( 2.72 \) | \( 2.718 \) |
| \( 0.9999 \) | \( 1.0 \) | \( 1.00 \) | \( 1.000 \) |
When rounding \( 0.9999 \) to 1 d.p., the answer is 1.0, not 1. The zero after the decimal point is required to show the degree of accuracy.
Significant figures count all digits that carry meaning, starting from the first non-zero digit.
| Number | Significant Figures | Round to 2 s.f. | Round to 3 s.f. |
|---|---|---|---|
| \( 45.678 \) | 5 | \( 46 \) | \( 45.7 \) |
| \( 0.00345 \) | 3 (\( 3,4,5 \)) | \( 0.0035 \) | \( 0.00345 \) |
| \( 1500 \) | 2 (\( 1,5 \)) | \( 1500 \) | \( 1500 \) |
| \( 0.05060 \) | 4 (\( 5,0,6,0 \)) | \( 0.051 \) | \( 0.0506 \) |
Significant Figures: "Start counting from the first non-zero digit, and keep going!"
Scientific notation expresses numbers in the form:
where \( 1 \le A < 10 \) and \( n \) is an integer.
| Ordinary Form | Scientific Notation | Reason |
|---|---|---|
| \( 3000 \) | \( 3 \times 10^3 \) | Decimal point moved 3 places left |
| \( 0.00056 \) | \( 5.6 \times 10^{-4} \) | Decimal point moved 4 places right |
| \( 6.02 \times 10^{23} \) | Already in scientific notation | — |
| \( 0.00123 \) | \( 1.23 \times 10^{-3} \) | Decimal point moved 3 places right |
Multiplication:
Division:
Addition/Subtraction: Convert to the same power of 10 first.
\( (3 \times 10^4) \times (2 \times 10^5) = 6 \times 10^9 \)
\( (6 \times 10^8) \div (2 \times 10^3) = 3 \times 10^5 \)
\( (2.5 \times 10^4) + (3.0 \times 10^4) = 5.5 \times 10^4 \)
When a measurement is approximate, we can calculate the error.
$$ \text{Absolute Error} = |\text{Exact Value} - \text{Approximate Value}| $$
$$ \text{Relative Error} = \frac{\text{Absolute Error}}{|\text{Exact Value}|} $$
$$ \text{Percentage Error} = \text{Relative Error} \times 100\% $$
DSE questions often give the exact value and approximate value and ask you to calculate the percentage error. Always use the exact value as the denominator in relative error.
Question: The exact value of a measurement is 50.0 cm. A student measures it as 49.5 cm. Find the absolute error, relative error, and percentage error.
Absolute Error = \( |50.0 - 49.5| = 0.5 \) cm
Relative Error = \( \frac{0.5}{50.0} = 0.01 \)
Percentage Error = \( 0.01 \times 100\% = 1\% \)
Question: Round \( 0.004567 \) to 2 significant figures.
The first two significant figures are \( 4 \) and \( 5 \).
The next digit is \( 6 \) (≥ 5), so round up.
Answer: \( 0.0046 \)
Question: Write \( 0.00000045 \) in scientific notation.
Move the decimal point 7 places to the right:
\( 0.00000045 = 4.5 \times 10^{-7} \)
Question: The exact value of a quantity is 250. A calculator gives 252. Find the percentage error.
Absolute Error = \( |250 - 252| = 2 \)
Relative Error = \( \frac{2}{250} = 0.008 \)
Percentage Error = \( 0.008 \times 100\% = 0.8\% \)
Question 1 MC
Round \( 0.009876 \) to 3 significant figures.
A. \( 0.00988 \) B. \( 0.00987 \) C. \( 0.0099 \) D. \( 0.01 \)
Question 2 MC
What is \( 3.2 \times 10^{-4} \) in ordinary form?
A. \( 0.00032 \) B. \( 0.0032 \) C. \( 0.032 \) D. \( 0.32 \)
Question 3 Short Answer
Calculate \( (4 \times 10^3) \times (5 \times 10^6) \). Express your answer in scientific notation.
Question 4 Short Answer
The exact value of a measurement is 80.0 m. A student estimates it as 79.2 m. Find the percentage error.
Question 5 MC
Which of the following is not a valid scientific notation?
A. \( 5.6 \times 10^3 \) B. \( 0.56 \times 10^4 \) C. \( 7.2 \times 10^{-2} \) D. \( 9.0 \times 10^0 \)
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An Exercise on Approximation & Errors Complete Guide | Scientific Notation + Significant Figures